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People seemed to enjoy my intuitive and visual explanation of Markov chain Monte Carlo so I thought it would be fun to do another one, this time focused on copulas.

If you ask a statistician what a copula is they might say “a copula is a multivariate distribution \(C(U_1, U_2, ...., U_n)\) such that marginalizing gives \(U_i \sim \operatorname{\sf Uniform}(0, 1)\) ”. OK… wait, what? I personally really dislike these math-only explanations that make many concepts appear way more difficult to understand than they actually are and copulas are a great example of that. The name alone always seemed pretty daunting to me. However, they are actually quite simple so we’re going to try and demistify them a bit. At the end, we will see what role copulas played in the 2007-2008 Financial Crisis.

Let’s start with an example problem case. Say we measure two variables that are non-normally distributed and correlated. For example, we look at various rivers and for every river we look at the maximum level of that river over a certain time-period. In addition, we also count how many months each river caused flooding. For the probability distribution of the maximum level of the river we can look to Extreme Value Theory which tells us that maximums are Gumbel distributed. How many times flooding occured will be modeled according to a Beta distribution which just tells us the probability of flooding to occur as a function of how many times flooding vs non-flooding occured.

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